Non-magnetic states - Boston University Physicsphysics.bu.edu/~sandvik/perimeter/l03.pdf ·...
Transcript of Non-magnetic states - Boston University Physicsphysics.bu.edu/~sandvik/perimeter/l03.pdf ·...
Non-magnetic states
For Jij>0 the ground state is the singlet;
The Néel states have higher energy (expectations; not eigenstates)
|!Naij ! = |"i#j!, |!Nb
ij ! = |#i"j!, $Hij! = %Jij/4
|!sij! =
|"i#j! $ |#i"j!%2
, Eij = $3Jij/4
The Néel states are product states; |!Naij ! = |"i#j! = |"i! $ |#j!
The singlet is a maximally entangled state (furthest from product state)
N>2: each spin tends to entangle with its neighbors (spins it interacts with)• entanglement is energetically favorable• but cannot singlet-pair with more than 1 spin• leads to fluctuating singlets (valence bonds)➡ less entanglement, <Hij> > -3Jij/4➡ closer to a product state (e.g., Néel)
Two spins, i and j, in isolation, Hij = Jij!Si · !Sj = Jij [Sz
i Szj + 1
2 (S+i S!j + S!i S+
j )]
• non-magnetic states possible (N=∞)➡ resonating valence-bond (RVB) spin liquid➡ valence-bond solid (VBS)
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Conditions on magnetic order: The Mermin-Wagner theoremA continuous symmetry cannot be broken for• a 2D system (classical or quantum-mechanical) at T>0• a 1D system at T=0,T>0
• quantum to classical mapping gives 2D T>0 system (path integral)
The Heisenberg model has a continuous symmetry• spin-rotation invariance [global SU(2) rotation invariance]• so cannot have Néel order at T>0 in 2D and not at all in 1D
1D Heisenberg chain (S = 1, 2, ...)• spin correlations decay exponentially at T=0 (the “Haldane conjecture”)
C(r) = !!Si · !Si+r" # ($1)re!r/!S , "S %& as S %&
1D Heisenberg chain (S = 1/2, 3/2, ...)• spin correlations decay algebraically (almost) at T=0
C(r) = !!Si · !Si+r" # ($1)r ln1/2(r/r0)r
, (T = 0)
2D Heisenberg model (e.g., square lattice)• spin correlation length diverges exponentially fast as T→0
C(rij) = !!Si · !Sj" # ($1)xij+yij e!rij/!, " %& as T % 0
Similar (even-odd n) behavior in n-leg S=1/2 spin ladders17Wednesday, April 7, 2010
Quantum phase transitions (T=0; change in ground-state)Example: Dimerized S=1/2 Heisenberg models• every spin belongs to a dimer (strongly-coupled pair)• many possibilities, e.g., bilayer, dimerized single layer
2D quantum spins map onto (2+1)D classical spins (Haldane) • Continuum field theory: nonlinear σ-model (Chakravarty, Halperin, Nelson)⇒ 3D classical Heisenberg (O3) universality class expected
Singlet formation on strong bonds ➙ Neel - disordered transitionGround state (T=0) phases
! ! �#$%& #s
weak interactions
strong interactions
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S=1/2 Heisenberg chain with frustrated interactions
= J2
= J1
• Antiferromagnetic “quasi order” (critical state) for g<0.2411... - exact solution - Bethe Ansatz - for J2=0 - bosonization (continuum field theory) approach gives further insights - spin-spin correlations decay as 1/r
- gapless spin excitations (“spinons”, not spin waves!)
C(r) = !!Si · !Si+r" # ($1)r ln1/2(r/r0)r
• VBS order for g>0.2411... the ground state is doubly-degenerate state - gap to spin excitations; exponentially decaying spin correlations
- singlet-product state is exact for g=0.5 (Majumdar-Gosh point)
C(r) = !!Si · !Si+r" # ($1)re!r/!
Different types of ground states, depending on the ratio g=J2/J1 (both >0)
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Frustration in higher dimensionsThere are many (quasi-)2D and 3D materials with geometric spin frustration• no classical spin configuration can minimize all bond energies
Kagome Pyrocloretriangular
(hexagonal)SrCu2(BO3)2
(Shastry-Sutherland)
A single triangular cell:• 6-fold degenerate Ising model• “120º Néel” order for vectors
Infinite triangular lattice• highly degenerate Ising model (no order)• “120º Néel” (3-sublattice) order for vectors
S=1/2 quantum triangular Heisenberg model• the classical 3-sublattice order most likely survives
[White and Chernyshev, PRL 2007]
S=1/2 Kagome system• very challenging, active research field; VBS or spin liquid?
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Frustration due to longer-range antiferromagnetic interactions in 2DQuantum phase transitions as some coupling (ratio) is varied• J1-J2 Heisenberg model is the prototypical example
H =!
!i,j"
Jij!Si · !Sj
= J1
= J2
g = J2/J1
• Ground states for small and large g are well understood‣ Standard Néel order up to g≈0.45‣ collinear magnetic order for g>0.6
0 ! g < 0.45 0.45 ! g < 0.6 ! 0" 6• A non-magnetic state exists between the magnetic phases‣ Most likely a VBS (what kind? Columnar or “plaquette?)‣ Some calculations (interpretations) suggest spin liquid
• 2D frustrated models are challenging ‣ no generally applicable unbiased methods (numerical or analytical)
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